Answer:
Step-by-step explanation:
x = cos θ + sin(10θ)
y = sin θ + cos(10θ)
Take derivative with respect to θ:
dx/dθ = -sin θ + 10 cos(10θ)
dy/dθ = cos θ - 10 sin(10θ)
Divide:
dy/dx = (dy/dθ) / (dx/dθ)
dy/dx = (cos θ - 10 sin(10θ)) / (-sin θ + 10 cos(10θ))
Evaluate the derivative at θ=0:
dy/dx = (cos 0 - 10 sin 0) / (-sin 0 + 10 cos 0)
dy/dx = 1/10
Evaluate the parametric functions at θ=0:
x = cos 0 + sin 0 = 1
y = sin 0 + cos 0 = 1
Writing the equation of the tangent line in point-slope form:
y - 1 = 1/10 (x - 1)
The equation of the tangent to the curve at the point corresponding to the given values of the parametric equations given is;
y - 1 = ¹/₁₀(x - 1)
We are given;
x = cos θ + sin(10θ)
y = sin θ + cos(10θ)
Since we want to find equation of tangent, let us first differentiate with respect to θ. Thus;
dx/dθ = -sin θ + 10cos (10θ)
Similarly;
dy/dθ = cos θ - 10sin(10θ)
To get the tangent dy/dx, we will divide dy/dθ by dx/dθ to get;
(dy/dθ)/(dx/dθ) = dy/dx = (cos θ - 10sin(10θ))/(-sin θ + 10cos(10θ))
To get the tangent, we will put the angle to be equal to zero.
Thus, at θ = 0, we have;
dy/dx = (cos 0 - 10sin 0)/(-sin 0 + 10cos 0)
dy/dx = 1/10
Also, at θ = 0, we can get the x-value and y-value of the parametric functions.
Thus;
x = cos 0 + sin 0
x = 1 + 0
x = 1
y = sin 0 + cos 0
y = 0 + 1
y = 1
Thus, the equation of the tangent line to the curve in point slope form gives us;
y - 1 = ¹/₁₀(x - 1)
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